I've done that too--I just eventually decided it was nicer to swap bowls, and let them thaw inside (no wet hands or wet gloves!)
I'm guessing 10 ounces would be big enough for one male chicken, in the winter, if you're giving fresh water at least twice a day (morning and afternoon/evening.)
But I don't know for sure.
I have seen estimates of 1 pint (16 ounces) per hen per day (laying hens, 24 hours.) A rooster probably needs less because he's not laying, but maybe more because he's bigger, but less if you're filling it more than once a day.
A larger dish should take longer to freeze, but I don't know if the time difference would be enough to matter.
You could measure how much your big rubber bowl holds, count how many chickens drank from it last winter, and see how much water per bird that was. If they did not drink it all each day, then obviously it was more than enough.
Or you could get some of each size and test both.
Sorry I don't have a good answer, just a bunch of "maybe" and "you could..."